Design and Data Analysis in Psychology II PRACTICE LESSON 2 School of Psychology

Susana Sanduvete Chaves Salvador Chacón Moscoso

DEPARTAMENTO DE PSICOLOGÍA EXPERIMENTAL

Exercise 1. A psychologist applies an Intelligence test to 10 pairs of monozygotic twins to verify if there are differences in intelligence (IT) between first-born and second-born twins. The IT scores of 10 pairs of twins appear below. IT1 and IT2 represent ‘IT’ scores twins’ born in the first and second place respectively. Assumptions were not accepted.

  pairs of monozygotic twins

IT1 116 100 110 100 124 134 128 110 100 130

IT2 109 89 98 110 115 110 105 90 121 105

Are there differences in intelligence (IT) between first-born and second-born twins? α=0.05

IT 1 IT 2 d Rank Sign116 109 7 1 +100 89 11 4 +110 98 12 5 +100 110 -10 3 -124 115 9 2 +134 110 24 9 +128 105 23 8 +110 90 20 6 +100 121 -21 7 -130 105 25 10 +

∑ R i:T +¿=1+ 4+5+2+9+8+6+10=45 ¿

T−¿=3+7=10¿

The smallest is T−¿=10 ¿

T c (α ,N )=T (0.05 ,10 )=8T=10>T c=8→H0

2 paired groupsQuantitative dependent variableWilcoxon T Test

Exercise 2. A psychologist applies an Intelligence test to 10 pairs of monozygotic twins to verify if there are differences in intelligence (IT) between first-born and second-born twins. The IT scores of 10 pairs of twins appear below. IT1 and IT2 represent ‘IT’ scores twins’ born in the first and second place respectively. Assumptions were not accepted.

  pairs of monozygotic twins

IT1 112 116 108 100 94 125 120 112 123 115

IT2 131 117 90 102 99 110 106 115 131 121

Are differences in intelligence (IT) between first-born and second-born twins?

IT 1 IT 2 d Rank Sign112 131 -19 10 -116 117 -1 1 -108 90 18 9 +100 102 -2 2 -94 99 -5 4 -125 110 15 8 +120 106 14 7 +112 115 -3 3 -123 131 -8 6 -115 121 -6 5 -

∑ R i:T +¿=9+8+7=24 ¿

T−¿=10+ 1+2+ 4+9+6+5=31 ¿

The smallest is T +¿=24 ¿

T c (α ,N )=T (0.05 ,10 )=8T=24>Tc=8→H0 rejected

2 paired groupsQuantitative dependent variableWilcoxon T Test

Exercise 3. We present the level of sociability in 10 pairs of brothers, one who practices team sports (a1) and the other who practices individual sports (a2). Assumptions were not accepted. Are there statistical differences in sociability between groups? (α = 0.05).

  Level of sociability

Team sports 125 115 130 140 140 115 140 125 140 135

Individual sports 110 122 125 120 140 124 123 137 135 145

TS IS d Rank Sign125 110 15 7 +115 122 -7 3 -130 125 5 1.5 +140 120 20 9 +140 140 0 0115 124 -9 4 -140 123 17 8 +125 137 -12 6 -140 135 5 1.5 +135 145 -10 5 -

∑ R i:T +¿=7+1.5+9+8+1.5=27¿

T−¿=3+4+6+5=18 ¿

The smallest is T−¿=18 ¿

T c (α ,N )=T (0.05 ,9 )=6T=18>T c=6→H0

2 paired groupsQuantitative dependent variableWilcoxon T Test

Only 9 because we don’t consider the “0” result

Exercise 4. A school psychologist considers that the academic background of students for university entrance is similar in private and public schools. Groups are considered independent. Assumptions were not accepted. To contrast the view, he gets the final average marks in an entrance test of 8 students from private schools and 10 public schools. Marks were the following:

Private schools: 3 3.2 4.8 5.5 5.8 6.5 7.5 8Public schools: 2 2.5 3 3.5 4.5 5 6 6.5 7 8.5

Can we admit the psychologist's hypothesis at a significance level of 5%?n1=8 Private

schoolsn2=10 Public

schools3.5 3 1 25 3.2 2 2.58 4.8 3.5 310 5.5 6 3.511 5.8 7 4.5

13.5 6.5 9 516 7.5 12 617 8 13.5 6.5

15 718 8.5

Suma R1 84 Suma R2 87

U=n1n2+n1(n1+1)

2−∑ R1=8·10+ 8 ·9

2−84=80+ 72

2−84=32

U=n1n2+n2 (n2+1 )

2−∑ R1=8 ·10+ 10 ·11

2−87=80+55−87=48

Smallest U = 32

U (α ,n2 ,n1 )=U ( 0.05 ,10,8)=17

U=32>U (0.05 , 10,8)=17→H 0

2 independent groupsQuantitative dependent variableMann-Whitney U

Exercise 5. We have anxiety measurements taken from 5 people with an endocrine problem and 4 healthy persons. Groups were considered independent. Assumptions were not accepted.

  Anxiety measurements  

Healthy 4.2 4.5 5.1 5.6   n1 = 4

Unhealthy 4.8 5.3 5.8 6.1 7.2 n2 = 5

Are there differences in the anxiety measurements between the unhealthy and healthy populations? (α = 0.05).

  Anxiety measurements ∑ R i  

Healthy 4.2 (1) 4.5 (2) 5.1 (4) 5.6 (6)   13 n1 = 4

Unhealthy 4.8 (3) 5.3 (5) 5.8 (7) 6.1 (8) 7.2 (9) 32 n2 = 5

U=n1n2+n1(n1+1)

2−∑ R1=4 ·5+ 4 ·5

2−13=20+10−13=17

U=n1n2+n2 (n2+1 )

2−∑ R1=4 ·5+ 5 ·6

2−32=20+15−32=3

Smallest U = 3

U (α ,n2 ,n1 )=U ( 0.05 ,5,4 )=1

U=3>U ( 0.05, 5,4 )=1→H 0

2 independent groupsQuantitative dependent variableMann-Whitney U

Exercise 6. We study the effects of humor on the human immune system. We obtain a random sample of 12 individuals and randomly assign 4 persons to one of the following three different experimental conditions. Viewing a: 1. humorous video; 2. emotionally stressful video; 3. emotionally neutral one. At the end of viewing, all people immunoglobulin ‘A’ amount in saliva was measured (index of body’s defenses). In the following table the values of immunoglobulin ‘A’ scores transformed into an ordinal scale are presented. The higher the value, the higher the bodies’ defenses are.

Experimental conditions; different video contentsHumorous Emotionally stressful Emotionally neutral

Immunoglobulin ‘A’ amount in

saliva

1 2 36 4 58 9 710 11 12

Are there statistical differences in bodies’ defenses between groups? α=0.05

Humorous Stressful Neutral

1 2 36 4 58 9 710 11 12

∑ R1=25 ∑ R2=26 ∑ R3=27

H= 12N (N+1)

·∑j=1

k R j2

n j−3 (N+1)

H= 1212 ·13 (252

4+ 262

4+272

4 )−3 ·13= 12156 ( 625

4+ 676

4+729

4 )−39= 12156

· 20304

−39=39.04−39=0.04

n1=n2=n3=4k=3α=0.05H c (α ,k , n1 , n2 , n3 )=5.692

H=0.04<H c=5.692→H 0

k groupsIndependent videosQuantitative dependent variableKurskal Wallis H

Exercise 7. We study the effects of humor on the human immune system. We obtain a random sample of 12 individuals and randomly assign 4 persons to one of the following three different experimental conditions. Viewing a: 1. humorous video; 2. emotionally stressful video; 3. emotionally neutral one. At the end of viewing, all people immunoglobulin ‘A’ amount in saliva was measured (index of body’s defenses). In the following table the values of immunoglobulin ‘A’ scores transformed into an ordinal scale are presented. The higher the value, the higher the bodies’ defenses are.

Experimental conditions; different video contentsHumorous Emotionally stressful Emotionally neutral

Immunoglobulin ‘A’ amount in

saliva

11 1 910 5 712 2 68 3 4

Are there statistical differences in bodies’ defenses between groups?

Humorous Stressful Neutral

11 1 910 5 712 2 68 3 4

∑ R1=41 ∑ R2=11 ∑ R3=26

H= 12N (N+1)

·∑j=1

k R j2

n j−3 (N+1)

H= 1212 ·13 (412

4+ 112

4+ 262

4 )−3 ·13= 12156 ( 1681

4+ 121

4+ 676

4 )−39= 12156

· 24784

−39=47.654−39=8.654

n1=n2=n3=4k=3α=0.05H c (α ,k , n1 , n2 , n3 )=5.692

H=8.654>H c=5.692→H 0rejected

k groupsIndependent videosQuantitative dependent variableKurskal Wallis H

Exercise 8. The following table presents data obtained in participants from different countries (A = France; B = Spain; and C = Portugal) referred to level of satisfaction with their economic situation. Assumptions were not accepted. Were there differences in level of satisfaction in participants from different countries? (α = 0.05).

A R B R C R

 6.4  6.8  7.2  8.3  8.4  9.1  9.4  9.7 

1112131718192021

 2.5  3.7  4.9  5.4  5.9  8.1  8.2 

23

5.58

1014

15.5

 1.3  4.1  4.9  5.2  5.5  8.2 

14

5.579

15.5

131 58 42

H= 12N (N+1)

·∑j=1

k R j2

n j−3 (N+1)

H= 1221 ·22 (1312

8+ 582

7+ 422

6 )−3·22= 12462 ( 17161

8+ 3368

7+ 422

6 )−39= 12462

·2920.268−66=75.927−66=9.927

n1=8n2=7n3=6k=3α=0.05H c (α , k , n1 , n2, n3 )=5.819

H=9.927>H c=5.819→H 0rejected

K independent groupsQuantitative dependent variableKurskal Wallis H

Exercise 9. The scores obtained by two groups of 10 students each one in an intelligence test, were the following ones:

A 40 46 35 17 11 22 20 31 18 22 262B 20 10 20 15 18 6 12 12 6 11 130

ΣAfter calculating F statistic, could you conclude that there are statistical differences between both groups? (α=0.05).n = 10y1 · j=26.2y2 · j=13

y1,2=39.2

2=19.6

SS SS df MS F sigBetween 871.2 1 871.2 11.367 0.003Within 1397.6 18 76.644Total 2250.8 19

SSB=n ( y · j− y · ·)2=10¿

SSW=∑ ( y · j− y · ·)2=¿

F(α ,k−1 , k (n−1))=F (0.05,1,18)=4.41F=11.367>F (0.05,1,18)=4.41→H0 rejected

Exercise 10. We have a random sample of people with high scores of anxiety. They are randomly distributed into three groups, so that n1 = 4, n2 = 3 and n3 = 2. We supply three different doses of a certain drug: 0.05 mg, 0.10 mg, and 0.20 mg. Under the assumption of independence of observations, normality of distributions and homogeneity of variance; do these doses influence state of anxiety? (α= 0.05). What can we conclude considering the significance and the effect size?

0.05 0.10 0.205 6 28 7 44 83

Σ 20 21 6

y1· j=204

=5

y2 · j=213

=7

y3 · j=62=3

y ··=5+7+3

9=5.22

SSB=n ( y · j− y · ·)2=4 ·(5−5.22)2+3 ·(7−5.22)2+2 ·(3−5.22)2=0.2+9.51+9.86=19.556

SSW=∑ ( y · j− y · ·)2=¿

df B=k−1=2df W=k (n−1 )=(n1−1 )+(n2−1 )+(n3−1 )=3+2+1=6df T=8SV SS df MS FB 19.556 2 9.778 3.259W 18 6 3T 27.556 8

F(α ,k−1 , k (n−1))=F (0.05 ,2,6)=5.14F=3.259<F(0.05 ,2,6)=5.14→H0

R2=SSBSST

=19.55637.556

=0.52¿

The non-significant effect can be due to low statistical power (increasing the sample size, we may obtain statistical differences).

Exercise 11. We have a random sample of 21 individuals, randomly distributed into three groups of 7 participants each. We want to test the effect of three forms of learning mathematical calculation (A, B, C) on performance. The results are the following:

A B C12

18 6

18

17 4

16

16 14

8 18 46 12 612

17 12

10

10 14

Σ 82

108 60

Are there differences between different learning methods? Can you specify between which concrete groups? Answer this last question using Tukey and Scheffé. Which statistic is the most conservative? Why? (α= 0.05)A)

y1· j=827

=11.71

y2 · j=108

7=15.43

y3 · j=607

=8.57

y ··=11.71+15.43+8.57

3=11.90

SSB=n ( y · j− y · ·)2=7· [(11.71−11.9 )2+(15.43−11.9)2+(8.57−11.9 )2 ]=164.952

SSW=∑ ( y · j− y · ·)2=¿

df B=k−1=2df W=k (n−1 )=3 ·6=18df T=20SV SS df MS FB 164.952 2 82.476 5.069W 292.857 18 16.27T 457.810 20

F(α ,k−1 , k (n−1))=F (0.05 ,2,18)=3.55F=5.069>F(0.05 ,2,6)=3.55→H 0rejected

B) Tukey:

Ψ (HDS )=qα, k (n−1 ), k √MSWn =q0.05,18, 3 √ 82.4767

=3.61√2.32≅ 5.5

|y1− y2|<HSD→3.72<5.5→H 0

|y1− y3|<HSD→3.14<5.5→H 0

|y2− y3|<HSD→6.86>5.5→H0 rejected

C) Scheffé:a1 a2 a3

+1 -1 0

Scheffé=√( k−1 ) · F(α , k−1 , k (n−1)) · MSW ·∑ a2

n=√2·5.069 ·16.27 · 2

7≅ 5.75

|y1− y2|<Scheffé→3.72<5.75→H 0

|y1− y3|<HSD→3.14<5.75→H 0

|y2− y3|<HSD→6.86>5.75→H 0 rejected

The most conservative statistic is Scheffé because is more difficult to reject the null hypothesis (5.75>5.5), and the empirical power has to be higher than the value of reference

Exercise 12. A randomized sample of 30 students from a school was randomly assigned to equal groups in order to study the effect of three different learning methods. The within groups sum of squares was 700; the mean of the first group was 30; the mean of the second group was 20; and the mean of the third group was 50. Calculate the ANOVA and conclude. Which are the concrete individual groups that present differences? Answer this last question using Tukey (α= 0.05)

N = 30, k = 3, n = 10.SSW=700y1=30y2=20y3=50y ··≅ 33.3SSB=n ( y · j− y · ·)

2=10· [ (30−33.3 )2+ (20−33.3 )2+ (50−33.3 )2 ]=10 · (11.09+177.69+277.89 )=4666.7SV SS df MS FB 4666.7 2 2333.35 90.91W 700 27 25.93T 5366.7 29

F(α ,k−1 , k (n−1))=F (0.05 ,2,27)=3.35F=90.91>F(0.05 ,2,27 )=3.35→H0 rejected

Tukey:

Ψ (HDS )=qα, k (n−1 ), k √MSWn =q0.05,27 ,3 √ 25.9310

=3.51√2.59≅ 5.65

|y1− y2|<HSD→10>5.65→H0rejected|y1− y3|<HSD→20>5.65→H 0 rejected|y2− y3|<HSD→30>5.65→H0 rejected

Exercise 13. Add contrasts 3 and 4 in order to obtain an orthogonal set of coefficients.

a1 a2 a3 a4 a5

Contrast 1 +3 0 -1 -1 -1Contrast 2 0 0 +1 -1 0Contrast 3 -1 4 -1 -1 -1Contrast 4 0 0 -1 -1 2

Exercise 14. One researcher would like to know if there are differences in math results when comparing urban schools (1 and 2) with rural schools (3 and 4), based on the data presented in the table below (α= 0.05):

School 1 School 2 School 3 School 48 5 9 47 4 7 35 6 6 56 5 7 6

Σ 26 (-1) 20 (-1) 29 (+1) 18 (+1)SSW = 16.75

y1=264

=6.5

y2=204

=5

y3=294

=7.25

y3=184

=4.5

SSBC=n(∑ a y j )

2

∑ a2 = 4 (−6.5−5+7.25+4.5 )2

4=0.252≅ 0.06

SV SS df MS FB 0.06 1 0.06 0.04W 16.75 12 1.4

F(α ,k−1 , k (n−1))=F (0.05 ,1,12)=4.75F=0.04<F ( 0.05,1,12)=4.75→H 0

Exercise 15. In a study, 6 non-orthogonal contrasts want to be carried out. Which is the risk that should be assumed for each contrast? (α = 0.05).

α c=αc=0.05

6=0.00833

α 1=0.00833 ;α 2=0.00833 ·2 ;α 3=0.00833 ·3; α4=0.00833 ·4 ;α 5=0.00833 ·5 ;α6=0.05

Exercise 16. A study of errors in recognition was carried out for four participants each exposed to target syllable in familiar, unfamiliar, and nonword-sound word types.

Word typeParticipant Familiar

wordUnfamiliar

wordNonword

soundΣ y i·

A 9 3 0 12 4B 6 2 1 9 3C 11 6 4 21 7D 10 5 3 18 6Σ 36 16 8

y · j 9 4 2Measure: number of errors.

Are there statistical differences in the number of errors depending on the word type? (α = 0.05).

y ··=204

=153

=5

SSB=k·∑ ( y i·− y · ·)2=4 · [(4−5)¿¿2+(3−5)2+(7−5)2+(6−5)2]=30¿SSS·A=∑ ( y ij+ y · ·− y i·− y · j )

2=¿

SSA=n∑( y · j− y ··)2=4 · [ (9−5 )2+ (4−5 )2+(2−5 )2 ]=4 ·26=104

df B=n−1=3df W=k−1=2

SV SS df MS FB 30 3A 104 2 52 77.6S·A 4 6 0.67

F(α ,k−1 , k (n−1))=F (0.05 ,2,9)=5.14F=77.6>F(0.05 ,2,6)=5.14→H0 rejected

Stage 2:

ε= 1k−1

=0.5

F=77.6>F(0.05 ,1,3)=10.13→H 0rejected

Exercise 17. After obtaining the results presented in the table below:ANOVA

DV

Sum of Squares df Mean Square F Sig.

Between Groups 112,000 2 56,000 2,955 ,074

Within Groups 398,000 21 18,952

Total 510,000 23

Conclude taking into account the significance and the effect size (α=0.05).

R2=SSBSST

=112510

≅ 0.22→low effect ¿¿

The effect probably doesn’t exist.